A note on functional derivatives on continuous paths

In this paper, we study the relation between Fréchet derivatives, Dupire derivatives in Dupire (2009) and the derivatives modified by Levental et al. (2010). After introducing the definition of Fréchet derivatives for non-anticipative functionals, we prove that the Dupire derivatives, the extended Fréchet derivatives and the new setup in Levental et al. (2010) are coherent on continuous paths.